@JeffLambert - I know what the rule is but my question is about how the rule would actually be enforced (unfortunately lame question editors always water down the issue). As I originally mentioned, there seemed to be nothing to question Ted Cruz's status as the PRESIDENTIAL nominee so I doubt that a Vice-President would be scrutinized more.
– Hannover FistJun 8 '16 at 20:19

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The requirement 'natural born citizen' has come to mean 'a citizen at the time of birth,' which qualifies Ted Cruz, but not Arnold. Attempting to compare the two and come to any conclusion is a non sequitur.
– Jeff LambertJun 8 '16 at 20:27

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"If Donald Trump were to announce Arnold Schwarzenegger" = the universe would fold in upon itself and that would be that.
– user1530Jun 9 '16 at 5:34

2 Answers
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There are a few places where Schwarzenegger could be kept from the office of VP (which he is definitely ineligible for).

The most likely place, of course, is the "accepting the VP slot" stage of the process. Schwarzenegger knows he's ineligible, and there's not much reason to think he'd try to run anyway.

The second place is disqualification from state ballots. State law normally requires that to be on a ballot, you must be eligible for the office. State boards of elections could take the initiative in refusing to put Schwarzenegger on the ballot, or the challenge-a-candidate process could be used to challenge his placement on the ballot. This process might involve the state courts, which are not bound by the same standing requirements as the federal courts (states get to mostly decide for themselves what sorts of cases their courts are allowed to hear). Between the first two stages, it's unlikely Schwarzenegger would make ballots as a VP candidate. Some people did try to do this for Cruz, but were shot down because their argument was wrong.

Next, you have the popular and electoral votes. Trump picking an illegal running mate could possibly turn away some voters (if he loses, then the practical importance of his VP pick is much lower).

Assuming he got this far, there is at least one case in US history where electoral votes were cast for a candidate who was, at the time the electors voted, unquestionably ineligible to become President. The candidate was Horace Greeley, and the year was 1872; he was ineligible because he died between the popular and electoral votes. While most of his electors then voted for someone else, a few cast their ballots for him. The House of Representatives passed a resolution that those votes not be counted, while the Senate passed a resolution that they should be counted; under the rules then in effect, that meant they weren't counted. Congress could potentially build on this precedent and assert authority over determinging whether presidential candidates meet the constitutional qualifications, like they already exercise over congressional candidates.

Such a decision would almost certainly be challenged, because nothing in the Constitution says Congress judges presidential qualifications (whereas it does say that Congress is the sole judge of its members' qualifications). But it's not an unreasonable position to take -- someone must be able to judge qualifications, and Congress (as an elected branch of government) is already entrusted with counting the votes. Also, Schwarzenegger is not qualified, so it's not like Congress unjustly said he wasn't a valid candidate. The outcome if the electoral votes are discarded would seem to be the Democratic VP nominee being the only one with valid electoral votes, and so the only one who can be voted for in the contingent election that follows the majority of votes being thrown out; they would win by default. But this is all somewhat speculative.

This is pretty clearly covered by the 12th Amendment of the Constitution.

no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States

Note that this does not prohibit anyone from campaigning to become Vice President. Even a non-citizen is allowed to say "vote for me!" under the 1st Amendment. It probably doesn't prohibit a political party from nominating whoever they want (or even a potted plant). But when a person submits official paperwork to put their name on the ballot, the relevant authority (e.g. Board of Elections) should reject ineligible candidates.

About the only way to set up a Supreme Court challenge would be to get some state secretary of state to refuse to certify electoral votes for the ineligible candidate. That would provide standing for the candidate and for opposing parties

This source seems to think that the courts might refuse to hear such a case until a candidate is actually on the ballot, or perhaps not even until they win election. But at that point the ruling would almost certainly go against them. Given that the Chief Justice administers the oath of office, it's highly unlikely that SCOTUS would decline jurisdiction.

That's nice but we also have something about the President having to be a "natural born Citizen." But evidently there is no set process to determine who is eligible since we had someone born in Canada running. The Supreme Court (they have the sour cream) would only review a case if someone could prove harm. Since the vice-president does almost nothing, it would be even harder to bring a case.
– Hannover FistJul 28 '16 at 21:41

Someone merely campaigning for nomination is several steps removed from actually taking office. As ruled in the Obama lawsuits, only an opposing candidate on the general election ballot (or a state secretary of state, responsible for the vote count) would have standing to challenge eligibility in court.
– Foo BarJul 28 '16 at 22:08